A SDFEM for system of two singularly perturbed problems of convection-diffusion type with discontinuous source term

TL;DR

This paper develops a finite element method using specialized meshes to accurately solve a system of two singularly perturbed convection-diffusion problems with discontinuous sources, capturing boundary and interior layers effectively.

Contribution

It introduces a finite element approach with Shishkin and Bakhvalov-Shishkin meshes for a coupled system of singularly perturbed problems, providing error estimates and numerical validation.

Findings

Error estimate of order O(N^{-1} log^{3/2} N) in energy norm

Numerical experiments confirm theoretical error bounds

Method effectively captures boundary and interior layers

Abstract

We consider a system of two singularly perturbed Boundary Value Problems (BVPs) of convection-diffusion type with discontinuous source terms and a small positive parameter multiplying the highest derivatives. Then their solutions exhibit boundary layers as well as weak interior layers. A numerical method based on finite element method (Shishkin and Bakhvalov-Shishkin meshes) is presented. We derive an error estimate of order $O(N^{-1}\ln^{3/2}{N})$ in the energy norm with respect to the perturbation parameter. Numerical experiments are also presented to support our theoretical results.